2 Background on non-abelian analogs of class groups

Let $Q$ be a global field and $\infty$ a place of $Q$ . In this paper, we are interested in the cases $Q=\mathbb{Q}$ or $\mathbb{F}_{q}(t)$ with the usual infinite place. For a separable, quadratic extension $K/Q$ , we let $K^{\operatorname{un},\infty }$ be the maximal unramified extension of $K$ that is split completely over all places of $K$ over $\infty$ , and let $G_{K}^{\operatorname{un},\infty }=\operatorname{Gal}(K^{\operatorname{un},\infty }/K)$ . We let $G_{K}$ be the maximal pro- $p$ quotient of $G_{K}^{\operatorname{un},\infty }$ .

Lemma 2.2. If $K/Q$ is a separable, quadratic extension, then all inertia subgroups of $\operatorname{Gal}(K^{\operatorname{un},\infty }/Q)$ and the decomposition group at infinity are contained in

$$\begin{eqnarray}\{1\}\cup \{r\in \operatorname{Gal}(K^{\operatorname{un},\infty }/Q)\setminus G_{K}^{\operatorname{un},\infty }\mid r^{2}=1\}.\end{eqnarray}$$

If $Q$ is a global field and $\infty$ is a place of $Q$ such that $Q$ has no non-trivial finite extensions unramified everywhere and split completely over $\infty$ (such as in our cases of interest $Q=\mathbb{Q}$ or $\mathbb{F}_{q}(t)$ ), we call $Q,\infty$ rational-like. Then we have that $\{r\in \operatorname{Gal}(K^{\operatorname{un},\infty }/Q)\setminus G_{K}^{\operatorname{un},\infty }\mid r^{2}=1\}$ is non-empty. So the exact sequence

$$\begin{eqnarray}1\rightarrow G_{K}^{\operatorname{un},\infty }\rightarrow \operatorname{Gal}(K^{\operatorname{un},\infty }/Q)\rightarrow \operatorname{Gal}(K/Q)\rightarrow 1\end{eqnarray}$$

splits. Any lift of the generator of $\operatorname{Gal}(K/Q)$ gives an order- $2$ automorphism of $G_{K}^{\operatorname{un},\infty }$ by conjugation.

In light of Proposition 2.3, we pick a lift $\unicode[STIX]{x1D70F}$ of the generator of $\operatorname{Gal}(K/Q)$ to $\operatorname{Gal}(K^{\operatorname{un},\infty }/Q)$ and let conjugation by $\unicode[STIX]{x1D70F}$ be our chosen generator-inverting automorphism $\unicode[STIX]{x1D70E}$ of $G_{K}^{\operatorname{un},\infty }$ . Further, the Schur–Zassenhaus theorem [Reference WilsonWil98, Proposition 2.3.3] guarantees that all the lifts of the generator of $\operatorname{Gal}(K/Q)$ to the pro- $p$ quotient $G_{K}$ of $\operatorname{Gal}(K^{\operatorname{un},\infty }/Q)$ (or the pro-odd quotient) are conjugate. Thus for an odd finite group $H$ with automorphism $\unicode[STIX]{x1D70E}$ , we then have that $|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)|$ does not depend on the choice of  $\unicode[STIX]{x1D70F}$ .

3 Boston–Bush–Hajir heuristics: background and notation

Koch and Venkov [Reference Koch and VenkovKV75] have shown that for an imaginary quadratic extension $K/\mathbb{Q}$ , the group $G_{K}$ satisfies certain properties we will now outline. For a pro- $p$ group $G$ , let $d(G):=\dim _{\mathbb{Z}/p\mathbb{Z}}H^{1}(G,\mathbb{Z}/p\mathbb{Z})$ and $r(G):=\dim _{\mathbb{Z}/p\mathbb{Z}}H^{2}(G,\mathbb{Z}/p\mathbb{Z})$ . These are, respectively, the generator rank and the relation rank of $G$ as a pro- $p$ group. For a pro-finite group $G$ , we define a GI-automorphism of $G$ to be a $\unicode[STIX]{x1D70E}\in \operatorname{Aut}(G)$ such that $\unicode[STIX]{x1D70E}$ acts as inversion on a set of (topological) generators. For a pro- $p$ group, this is equivalent to requiring that $\unicode[STIX]{x1D70E}^{2}=1$ , which $\unicode[STIX]{x1D70E}$ are called involutions, and $\unicode[STIX]{x1D70E}$ acts as inversion on the abelianization of  $G$  [Reference BostonBos91].

Koch and Venkov [Reference Koch and VenkovKV75] have shown that for an imaginary quadratic extension $K/\mathbb{Q}$ , the group $G_{K}$ is a Schur- $\unicode[STIX]{x1D70E}$ group. The groups $G_{K}$ we are considering in the function field case are also Schur- $\unicode[STIX]{x1D70E}$ groups when $p\nmid q-1$ . This follows by class field theory, Proposition 2.3 above, and the upper bound on $r(G_{K})-d(G_{K})$ , namely $0$ , due to Shafarevich, given as [Reference Hajir, Maire and CasacubertaHM01, Theorem 2.2]. Note that $r(G_{K})-d(G_{K})\geqslant 0$ since $G_{K}^{\text{ab}}$ is finite and so the upper bound of 0 yields $r(G_{K})-d(G_{K})=0$ .

We will put a measure on the set of isomorphism classes of Schur $\unicode[STIX]{x1D70E}$ -groups in order to state the Boston–Bush–Hajir heuristics. For this, we first need to define a $\unicode[STIX]{x1D70E}$ -algebra (in the sense of measure theory – not our automorphism  $\unicode[STIX]{x1D70E}$ ) on this set. Since many infinite Schur $\unicode[STIX]{x1D70E}$ -groups are expected to occur as $G_{K}$ with density 0, it makes sense to focus on certain finite quotients of these groups.

Any pro- $p$ group $G$ has a lower $p$ -central series defined as $P_{0}(G):=G$ and for $n\geqslant 0$ , we let $P_{n+1}(G)$ be the closed subgroup generated by $[G,P_{n}(G)]$ and $P_{n}(G)^{p}$ . The groups $P_{0}(G)\geqslant P_{1}(G)\geqslant P_{2}(G)\geqslant \cdots$ form a descending chain of characteristic subgroups of $G$ called the lower $p$ -central series. The $p$ -class of a finite $p$ -group $G$ is the smallest $c\geqslant 0$ for which $P_{c}(G)=\{1\}$ . Note that for a finitely generated pro- $p$ group $G$ , the successive quotients $P_{n}(G)/P_{n+1}(G)$ are finite abelian groups of exponent $p$ , and so, in particular, if $P_{c}(G)=\{1\}$ , then $G$ must be finite. The lower $p$ -central series and $p$ -class can be thought of as analogous to the lower central series and nilpotency class, respectively. Note that $P_{1}(G)$ is also the Frattini subgroup  $\unicode[STIX]{x1D6F7}(G)$ .

For a pro- $p$ group $G$ , we define $Q_{c}(G):=G/P_{c}(G)$ , the maximal quotient of $G$ with $p$ -class at most $c$ . So $Q_{c}(G_{K})$ is the Galois group of the maximal unramified $p$ -extension of $K$ among extensions of Galois group with $p$ -class at most $c$ . Note that since a Schur $\unicode[STIX]{x1D70E}$ -group $G$ (such as $G_{K}$ ) is finitely generated, we have that $Q_{c}(G)$ is finite. It may be that $Q_{c}(G)$ has $p$ -class strictly less than $c$ : certainly when $G$ itself has $p$ -class strictly less than $c$ , this happens, but in fact since the subquotients of the lower $p$ -central series for $G$ and for $Q_{c}(G)$ are the same up to index $c$ , this is the only way it can happen.

Let $\unicode[STIX]{x1D6FA}$ be the $\unicode[STIX]{x1D70E}$ -algebra on the set of isomorphism classes of Schur $\unicode[STIX]{x1D70E}$ -groups generated by the sets

(2) $$\begin{eqnarray}\{G\mid Q_{c}(G)\simeq P\}\end{eqnarray}$$

for each finite $p$ -group $P$ and fixed $c$ . For example, we can fix a Schur $\unicode[STIX]{x1D70E}$ -group $G_{0}$ and take the intersection over all $c$ of $\{G\mid Q_{c}(G)\simeq Q_{c}(G_{0})\}$ to see that $\unicode[STIX]{x1D6FA}$ contains the singleton set containing the class of  $G_{0}$ .

We will next define a measure on the set of isomorphism classes of Schur $\unicode[STIX]{x1D70E}$ -groups for a $\unicode[STIX]{x1D70E}$ -algebra containing $\unicode[STIX]{x1D6FA}$ . Any Schur $\unicode[STIX]{x1D70E}$ -group of rank $g$ can be presented as a quotient of the free pro- $p$ group $F_{g}$ on $g$ generators $x_{1},\ldots ,x_{g}$ (with GI-automorphism $\unicode[STIX]{x1D70E}(x_{i})=x_{i}^{-1}$ ) by $g$ relations chosen from $X=\{s\in \unicode[STIX]{x1D6F7}(F_{g})\mid \unicode[STIX]{x1D70E}(s)=s^{-1}\}$ . Since $X$ is a closed subset of the profinite group $F_{g}$ , we have a natural profinite probability measure $\unicode[STIX]{x1D707}$ on $X$ from the limit of the uniform measures on finite quotients of $F_{g}$ , on the $\unicode[STIX]{x1D70E}$ -algebra generated by fibers of these quotients.

The Boston–Bush–Hajir probability measure $\unicode[STIX]{x1D707}_{\text{BBH}}$ will be given by randomly selecting such relations. However, this only gives a measure for a fixed rank $g$ of Schur $\unicode[STIX]{x1D70E}$ -groups. Since, however, the rank of a Schur $\unicode[STIX]{x1D70E}$ -group is the rank of its abelianization (in fact, of the quotient of the abelianization $G/\unicode[STIX]{x1D6F7}(G)$ , by the Burnside basis theorem), we can use the Cohen–Lenstra heuristics to predict how often each rank $g$ occurs. Let

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{\text{CL}}(g):=\mathop{\sum }_{G~\text{fin. ab.},~\text{rk}~g~p\text{-}\text{gp}}\unicode[STIX]{x1D707}_{\text{CL}}(G)=p^{-g^{2}}\mathop{\prod }_{k=1}^{g}(1-p^{-k})^{-2}\mathop{\prod }_{i=1}^{\infty }(1-p^{-i}).\end{eqnarray}$$

The above formula is from [Reference Cohen and LenstraCL84, Theorem 6.3]. Let $A$ be a set of isomorphism classes of rank $g$ Schur $\unicode[STIX]{x1D70E}$ -groups. Then we define

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{\text{BBH}}(A):=\unicode[STIX]{x1D707}_{\text{CL}}(g)\unicode[STIX]{x1D707}(\{(r_{1},\ldots ,r_{g})\in X^{g}\mid F_{g}/\langle \langle r_{1},\ldots ,r_{g}\rangle \rangle \in A\}),\end{eqnarray}$$

whenever $\{(r_{1},\ldots ,r_{g})\in X^{g}\mid F_{g}/\langle \langle r_{1},\ldots ,r_{g}\rangle \rangle \in A\}$ is measurable, where the double angle brackets denote the closed normal subgroup generated by the elements. We can think of this measure as generating a random group by picking a rank $g$ according to the Cohen–Lenstra measure and then independently creating a random Schur $\unicode[STIX]{x1D70E}$ -group of rank $g$ by taking the quotient of the free pro- $p$ group $F_{g}$ on $g$ generators by $g$ randomly chosen relations in  $X$ . Note that this process does not necessarily produce a Schur $\unicode[STIX]{x1D70E}$ -group, as there may be redundancy among the relations and so the resulting group may not have relation rank  $g$ . However, such redundancy happens with probability 0 (the abelianization would be infinite, and, as noted by Friedman and Washington [Reference Friedman and WashingtonFW89], this occurs with zero probability under $\unicode[STIX]{x1D707}_{\text{CL}}$ , which is induced on abelianizations from $\unicode[STIX]{x1D707}_{\text{BBH}}$ [Reference Boston, Bush and HajirBBH16, Theorem 2.20]).

Let $X_{c}=\{s\in \unicode[STIX]{x1D6F7}(Q_{c}(F_{g}))\mid \unicode[STIX]{x1D70E}(s)=s^{-1}\}$ . Note that $X_{c}$ is a finite set and has a uniform discrete probability measure $\unicode[STIX]{x1D707}_{c}$ that pulls back to $\unicode[STIX]{x1D707}$ on $X$ . If $P$ is a fixed finite $p$ -group with $d(P)=g$ , we define $\unicode[STIX]{x1D707}_{\text{BBH},c}(P):=\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq P\})$ , and then

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{\text{BBH},c}(P)=\unicode[STIX]{x1D707}_{\text{CL}}(g)\unicode[STIX]{x1D707}_{c}(\{(r_{1},\ldots ,r_{g})\in X_{c}^{g}\mid Q_{c}(F_{g})/\langle \langle r_{1},\ldots ,r_{g}\rangle \rangle \simeq P\}).\end{eqnarray}$$

In particular $\{G\mid Q_{c}(G)\simeq P\}$ is measurable for $\unicode[STIX]{x1D707}_{\text{BBH}}$ .

If $P\simeq Q_{c}(G)$ for some Schur $\unicode[STIX]{x1D70E}$ -group $G$ , we call $P$ a Schur $\unicode[STIX]{x1D70E}$ -ancestor group. Note that a Schur $\unicode[STIX]{x1D70E}$ -ancestor group is necessarily a finite $p$ -group with a GI-automorphism (though these conditions are not sufficient). The Schur $\unicode[STIX]{x1D70E}$ -ancestor groups are exactly those presented as $Q_{c}(F_{g})/\langle \langle r_{1},\ldots ,r_{g}\rangle \rangle$ for some $r_{1},\ldots ,r_{g}\in X_{c}$ . This is because one can choose an irredundant lift of the relations from $X_{c}$ to $X$ to give a Schur $\unicode[STIX]{x1D70E}$ -group [Reference Boston, Bush and HajirBBH16]. In particular, for any Schur $\unicode[STIX]{x1D70E}$ -ancestor group $G$ of $p$ -class $c$ , we have that $\unicode[STIX]{x1D707}_{\text{BBH},c}(G)>0.$

4 Boston–Bush–Hajir moments

We now determine the moments of the measure $\unicode[STIX]{x1D707}_{\text{BBH}}$ as stated in Theorem 1.3.

Theorem 4.1 (Moments of $\unicode[STIX]{x1D707}_{\text{BBH}}$ ).

Let $H$ be a finite $p$ -group of $p$ -class $c$ with a GI-automorphism $\unicode[STIX]{x1D70E}$ . Then

$$\begin{eqnarray}\int _{G}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G,H)|\,d\unicode[STIX]{x1D707}_{\text{BBH}}=\mathop{\sum }_{G~\mathit{Schur}~\unicode[STIX]{x1D70E}\text{-}\mathit{ancestor~of}~p\text{-}\mathit{class}~c}\unicode[STIX]{x1D707}_{\text{BBH},c}(G)|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G,H)|=1.\end{eqnarray}$$

Note the hypothesis that $\unicode[STIX]{x1D70E}$ is $GI$ on $H$ does not place any real restriction, because if we have a surjection $G\rightarrow H$ that takes a GI-automorphism $\unicode[STIX]{x1D70E}_{G}$ on $G$ to any automorphism $\unicode[STIX]{x1D70E}_{H}$ on $H$ , then $\unicode[STIX]{x1D70E}_{H}$ must also be GI.

Let $H$ be a finite $p$ group with an order- $2$ automorphism $\unicode[STIX]{x1D70E}$ . We write

$$\begin{eqnarray}Z(H)=\{g\in H\mid \unicode[STIX]{x1D70E}(g)=g\}\end{eqnarray}$$

and $Y(H)=\{g\in H\mid \unicode[STIX]{x1D70E}(g)=g^{-1}\}$ . This notation implicitly depends on $\unicode[STIX]{x1D70E}$ . We now prove several lemmas that will be used in the proof of Theorem 4.1.

Proof. Associated to the exact sequence $1\rightarrow \ker (\unicode[STIX]{x1D719})\rightarrow G\rightarrow H\rightarrow 1$ is the exact sequence

$$\begin{eqnarray}\cdots \rightarrow H^{0}(\langle \unicode[STIX]{x1D70E}\rangle ,G)\rightarrow H^{0}(\langle \unicode[STIX]{x1D70E}\rangle ,H)\rightarrow H^{1}(\langle \unicode[STIX]{x1D70E}\rangle ,\ker (\unicode[STIX]{x1D719}))\rightarrow \cdots \,.\end{eqnarray}$$

The first and second terms are $Z(G)$ and $Z(H)$ respectively. The last term is $H^{1}(\langle \unicode[STIX]{x1D70E}\rangle ,\ker (\unicode[STIX]{x1D719}))$ , which vanishes by the Schur–Zassenhaus theorem since $p$ is odd.◻

Proof. The first two claims are clear. Using the above two lemmas, we then observe

$$\begin{eqnarray}|Y(K)|=\frac{|K|}{|Z(K)|}=\frac{|G|/|H|}{|Z(G)|/|Z(H)|}=\frac{|Y(G)|}{|Y(H)|},\end{eqnarray}$$

which proves the final claim. ◻

Lemma 4.6. Let $H$ be a finite $p$ -group of generator rank $r$ with a GI-automorphism $\unicode[STIX]{x1D70E}$ . Then

$$\begin{eqnarray}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(F_{d},H)|=\frac{|Y(H)|^{d}(p^{d}-p^{r-1})\cdots (p^{d}-1)}{p^{dr}}.\end{eqnarray}$$

Proof of Theorem 4.1.

Since a surjection from $G$ to $H$ factors through $Q_{c}(G)$ , we see that $f(G)=|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G,H)|$ is in fact a measurable function and that the first equality is by definition of the two measures.

Let $H$ have generator rank $r$ . The random group $G$ is constructed first by picking a random generator rank $d$ for $G$ according to the Cohen–Lenstra measure, and then taking a random quotient of $F_{d}$ . Certainly, any surjection $G\rightarrow H$ lifts uniquely to a surjection $F_{d}\rightarrow H$ . From Lemma 4.6 we see there are $|Y(H)|^{d}(p^{d}-p^{r-1})\cdots (p^{d}-1)/p^{dr}$   $\unicode[STIX]{x1D70E}$ -equivariant surjections $F_{d}\rightarrow H$ . A surjection $\unicode[STIX]{x1D719}:F_{d}\rightarrow H$ factors through $G$ if and only if the $d$ random relations in $Y(\unicode[STIX]{x1D6F7}(F_{d}))$ that present $G$ are in $\ker (\unicode[STIX]{x1D719})$ , the probability of which we now compute. Since $H$ is $p$ -class $c$ , we may equivalently take the random relations in $Y(\unicode[STIX]{x1D6F7}(F_{d})/P_{c}(F_{d}))$ .

Let $F:=F_{d}/P_{c}(F_{d})$ . The probability that a random relation in $X_{c}=Y(\unicode[STIX]{x1D6F7}(F))$ is in $\ker (\unicode[STIX]{x1D719})$ is $|\text{ker}(\unicode[STIX]{x1D719})\cap Y(\unicode[STIX]{x1D6F7}(F))|/|Y(\unicode[STIX]{x1D6F7}(F))|$ . Applying Lemma 4.4 to the surjection $\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6F7}(F)\rightarrow \unicode[STIX]{x1D6F7}(G)$ , we see that $|\text{ker}(\unicode[STIX]{x1D719})\cap Y(\unicode[STIX]{x1D6F7}(F))|/|Y(\unicode[STIX]{x1D6F7}(F))|=|Y(\unicode[STIX]{x1D6F7}(G))|^{-1}$ . Also, applying Lemma 4.4 to the quotient $G\rightarrow G/\unicode[STIX]{x1D6F7}(G)$ , we have that $|Y(\unicode[STIX]{x1D6F7}(G))|=|Y(G)|/p^{r}$ , since $\unicode[STIX]{x1D70E}$ acts on all of $G/\unicode[STIX]{x1D6F7}(G)$ by inversion. Thus, the probability that $d$ random relations are in $\ker (\unicode[STIX]{x1D719})$ , and so the map $\unicode[STIX]{x1D719}$ factors through the random $G$ , is $p^{dr}/|Y(H)|^{d}$ .

Multiplying by the number of $\unicode[STIX]{x1D70E}$ -equivariant surjections $F_{d}\rightarrow H$ , we find that among generator rank $d$ groups $G$ , the expected number of $\unicode[STIX]{x1D70E}$ -equivariant surjections to $H$ is $(p^{d}-p^{r-1})\cdots (p^{d}-1)$ , which is the number of surjections from a rank $d$ abelian $p$ -group to $(\mathbb{Z}/p\mathbb{Z})^{r}$ . Thus the expected number of $\unicode[STIX]{x1D70E}$ -equivariant surjections is

$$\begin{eqnarray}\mathop{\sum }_{d\geqslant 0}\unicode[STIX]{x1D707}_{\text{CL}}(d)(p^{d}-p^{r-1})\cdots (p^{d}-1)=\mathop{\sum }_{A}\unicode[STIX]{x1D707}_{\text{CL}}(A)|\!\operatorname{Sur}(A,(\mathbb{Z}/p\mathbb{Z})^{r})|=1,\end{eqnarray}$$

by the moments formula for the Cohen–Lenstra measure. ◻

In fact, we will see in Theorem 4.9 that the moments where $H$ is a Schur $\unicode[STIX]{x1D70E}$ -ancestor group characterize $\unicode[STIX]{x1D707}_{\text{BBH}}$ as a measure on $\unicode[STIX]{x1D6FA}$ . At each $p$ -class, showing the moments characterize the measure amounts to inverting an infinite-dimensional matrix. Our method to invert this matrix can be seen as a generalization of the method of [Reference Ellenberg, Venkatesh and WesterlandEVW16, Lemma 8.2], which proves that the moments characterize the Cohen–Lenstra measure on finite abelian $p$ -groups. First we need an infinite-dimensional linear algebra lemma, since our infinite matrices are not quite as simple as those in [Reference Ellenberg, Venkatesh and WesterlandEVW16, Lemma 8.2].

Lemma 4.7. Let $a_{i,j}$ be non-negative real numbers indexed by pairs of natural numbers $i,j$ , such that for all $i$ we have $a_{i,i}=1$ , and also $\sup _{i}\sum _{j}a_{ij}<2$ . Let $x_{j},y_{j}$ be non-negative reals indexed by natural numbers  $j$ . If for all  $i$ ,

$$\begin{eqnarray}\mathop{\sum }_{j}a_{i,j}x_{j}=\mathop{\sum }_{j}a_{i,j}y_{j}=1,\end{eqnarray}$$

then $x_{j}=y_{j}$ for all $j$ .

Next, we will prove a formula for $\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq P\})$ for a given Schur $\unicode[STIX]{x1D70E}$ -ancestor group $P$ . The formula combines [Reference Boston, Bush and HajirBBH16, Theorems 2.25 and 2.29], which are subject to a further conjecture called KIP, but we prove below that the combined formula is not conjectural. For the formula, we will need one further invariant of $p$ -groups. For a finite $p$ -group $G$ of $p$ -class $c$ presented as $F/R$ , where $F$ is a free group of $d(G)$ generators, then $h(G)$ is defined to be the dimension of the quotient of $R$ by the topological closure of the subgroup $R^{p}[F,R]P_{c}(F)$ (by [Reference O’BrienO’B90] and [Reference Boston, Bush and HajirBBH16, Remark 2.4] the quantity does not depend on the choice of presentation).

Alternatively, the $p$ -groups of $p$ -class ${\leqslant}c$ form a variety of groups whose free objects are precisely the groups $Q_{c}(F_{d})$ . For a group $G$ in this variety, we can let $h_{c}(G)$ be the number of relators required to present $G$ in this variety. If $G$ is $p$ -class $c$ , then $h_{c}(G)=h(G)$ and if $G$ is $p$ -class smaller than $c$ , then $h_{c}(G)=r(G)$ .

Lemma 4.8. Fix a $c$ . Let $g=d(G)$ and $h=h_{c}(G)$ . We have

$$\begin{eqnarray}\frac{\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq P\})}{\unicode[STIX]{x1D707}_{\text{CL}}(g)}=\frac{p^{g^{2}}}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(G)|}\mathop{\prod }_{k=1}^{g}(1-p^{-k})\mathop{\prod }_{k=1+g-h}^{g}(1-p^{-k}).\end{eqnarray}$$

Proof. Let $F_{c}=Q_{c}(F_{g})$ . We need to compute the sum of the probabilities that a given $g$ -tuple of relations $v\in X_{c}^{g}$ generates $\overline{R}$ as a normal subgroup of $F_{c}$ , where $\overline{R}$ runs over all normal subgroups of $F_{c}$ with quotient $G$ . The key thing to note here is that since each element of $X_{c}$ is inverted by $\unicode[STIX]{x1D70E}$ , any subgroup generated by elements of $X_{c}$ is $\unicode[STIX]{x1D70E}$ -invariant, as is the normal closure of such a subgroup. Thus if $\overline{R}$ is a normal subgroup of $F_{c}$ that is not $\unicode[STIX]{x1D70E}$ -invariant, then the probability that is generated as a normal subgroup by relations from $X_{c}$ is $0$ . In [Reference Boston, Bush and HajirBBH16], the conjectural property KIP (kernel invariance property) was assumed to ensure that every normal subgroup with quotient $G$ is $\unicode[STIX]{x1D70E}$ -invariant. We do not assume this, since by the above remark we can restrict our attention to the set of $\unicode[STIX]{x1D70E}$ -invariant normal subgroups with quotient  $G$ .

The number of $\unicode[STIX]{x1D70E}$ -invariant normal subgroups of $F_{c}$ with quotient $G$ is $|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(F_{c},G)|/|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(G)|$ , by counting the quotient maps and dividing by how often maps give isomorphic quotients. (There are similarly $|\!\operatorname{Sur}(F_{c},G)|/|\!\operatorname{Aut}(G)|$ normal subgroups with quotient  $G$ , but if there are any that are not $\unicode[STIX]{x1D70E}$ -invariant we have already seen they have $0$ probability of being generated by our relations in  $X_{g}$ .) The probability that a $g$ -tuple of relations $v\in X_{c}^{g}$ generates a $\unicode[STIX]{x1D70E}$ -invariant $\overline{R}$ as a normal subgroup can be computed by the earlier methods of [Reference Boston, Bush and HajirBBH16]. We give a slightly alternative treatment here.

First note that by Lemma 4.6, $|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(F_{c},G)|=|Y(G)|^{g}\prod _{k=1}^{g}(1-p^{-k})$ , since every such surjection from the free pro- $p$ group $F_{g}$ on $g$ generators factors through $F_{c}$ . As for the probability that $v\in X_{c}^{g}$ normally generates $\overline{R}$ , this happens if and only if its image generates the $\mathbb{F}_{p}$ -vector space $V=\overline{R}/\overline{R}^{\ast }$ , where $R$ is the preimage of $\overline{R}$ in $F_{g}$ , $R^{\ast }$ is the topological closure of $R^{p}[F_{g},R]$ , and $\overline{R}^{\ast }=P_{c}(F_{g})R^{\ast }/P_{c}(F_{g})$ [Reference GruenbergGru76, Proposition 2.8]. When $G$ is $p$ -class $c$ , the dimension of $V$ is $h$ (by definition of  $h$ ). When $G$ is $p$ -class ${<}c$ , we have $P_{c-1}(F_{g})\subset R$ and so $P_{c}(F_{g})$ is a subgroup of $R^{\ast }$ . Then $V=R/R^{\ast }$ , which has dimension $r(G)$ . Let $s=\dim V$ , which we have just determined in each case. The number of $g$ -tuples generating $V$ is $\prod _{k=1}^{s}(p^{g}-p^{s-k})$ and so we just need the size of the intersection of $X_{c}$ with a fiber of the quotient map $r:\overline{R}\rightarrow V$ .

We claim each of these has $|\overline{R}^{\ast }|/|Z(\overline{R})|$ elements. This follows by considering the map $f$ of Lemma 4.5, defined by $f(g)=g^{-1}\unicode[STIX]{x1D70E}(g)$ . Since $V$ is abelian, $f\circ r=-2r$ , whose fibers have the same size as those of $r$ , namely $|\overline{R}^{\ast }|$ , since $p$ is odd. On the other hand, $f\circ r=r\circ f$ , the size of the fibers of which are the size of those of $r$ times those of  $f$ . This latter term is $|Z(\overline{R})|$ by Lemma 4.2. Putting these facts together establishes the claim.

To recap, the desired measure is the sum over $|Y(G)|^{g}\prod _{k=1}^{g}(1-p^{-k})/|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(G)|$ terms of the number of $v$ in $X_{c}^{g}$ normally generating each $\overline{R}$ , which we just found to be $\prod _{k=1}^{s}(p^{g}-p^{s-k})(|\overline{R}^{\ast }|/|Z(\overline{R})|)^{g}$ , divided by the total number of $v$ , namely $|X_{c}|^{g}$ . In other words,

$$\begin{eqnarray}\mathop{\prod }_{k=1}^{s}(p^{g}-p^{s-k})\mathop{\prod }_{k=1}^{g}(1-p^{-k})\frac{(|\overline{R}^{\ast }|/|Z(\overline{R})|)^{g}|Y(G)|^{g}}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(G)|\,|X_{c}|^{g}}.\end{eqnarray}$$

It remains to show that $|Y(G)|\,|\overline{R}^{\ast }|/(|Z(\overline{R})|\,|X_{c}|)=p^{g-s}$ . This follows from Lemma 4.4, which says that $|Y(F_{c})|=|Y(G)|\,|Y(\overline{R})|$ and $|Y(F_{c})|=|Y(\unicode[STIX]{x1D6F7}(F_{c}))|\,|Y(F_{c}/\unicode[STIX]{x1D6F7}(F_{c}))|=|X_{c}|p^{g}$ . Thus, $|X_{c}|=|Y(G)|\,|Y(\overline{R})|p^{-g}$ . Combining this with $|Y(\overline{R})|\,|Z(\overline{R})|=|\overline{R}|$ (Lemma 4.2) and $|\overline{R}|/|\overline{R}^{\ast }|=p^{s}$ gives the result.◻

Theorem 4.9 (Moments characterize $\unicode[STIX]{x1D707}_{\text{BBH}}$ ).

Let $\unicode[STIX]{x1D708}$ be a measure on $\unicode[STIX]{x1D6FA}$ such that for every Schur $\unicode[STIX]{x1D70E}$ -ancestor group $H$ ,

$$\begin{eqnarray}\int _{G}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G,H)|\,d\unicode[STIX]{x1D708}=1.\end{eqnarray}$$

Then $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D707}_{\text{BBH}}$ .

Note that Schur $\unicode[STIX]{x1D70E}$ -ancestor groups are a proper subset of finite $p$ -groups with GI-automorphisms, so this theorem does not require all of the moments determined in Theorem 4.1.

Proof. By Carathéodory’s theorem, a measure $\unicode[STIX]{x1D708}$ on $\unicode[STIX]{x1D6FA}$ is determined by the measures $\unicode[STIX]{x1D708}(\{G\mid Q_{c}(G)\simeq S\})$ for all Schur $\unicode[STIX]{x1D70E}$ -ancestor groups $S$ . If $G$ is a Schur $\unicode[STIX]{x1D70E}$ -group, then $Q_{c}(G)$ is either a Schur $\unicode[STIX]{x1D70E}$ -ancestor group of $p$ -class $c$ or a Schur $\unicode[STIX]{x1D70E}$ -group of $p$ -class ${<}c$ . (This is because if $Q_{c}(G)$ is $p$ -class ${<}c$ then $Q_{c}(G)=G$ .) Let ${\mathcal{S}}$ be the set of isomorphism classes of groups that are either a Schur $\unicode[STIX]{x1D70E}$ -ancestor group of $p$ -class $c$ or a Schur $\unicode[STIX]{x1D70E}$ -group of $p$ -class ${<}c$ .

For $H$ a Schur $\unicode[STIX]{x1D70E}$ -ancestor group of $p$ -class $c$ , we have that

$$\begin{eqnarray}\mathop{\sum }_{S\in {\mathcal{S}}}\unicode[STIX]{x1D708}(\{G\mid Q_{c}(G)\simeq S\})|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(S,H)|=1\end{eqnarray}$$

and

$$\begin{eqnarray}\mathop{\sum }_{S\in {\mathcal{S}}}\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq S\})|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(S,H)|=1.\end{eqnarray}$$

We can index ${\mathcal{S}}$ by natural numbers $S_{1},S_{2},\ldots \,$ . We then apply Lemma 4.7 with

$$\begin{eqnarray}a_{i,j}=\frac{|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(S_{j},S_{i})|}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|}\end{eqnarray}$$

and $x_{j}=\unicode[STIX]{x1D708}(\{G\mid Q_{c}(G)\simeq S_{j}\})|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|$ and $y_{j}=\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq S_{j}\})|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|$ , which will prove the proposition. We must verify that $\sum _{j}a_{i,j}<2$ .

Using the explicit formulae for $\unicode[STIX]{x1D707}_{\text{CL}}(d)$ (from [Reference Cohen and LenstraCL84]) and for $\unicode[STIX]{x1D707}_{\text{BBH}}$ (from Lemma 4.8), we have that

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq S_{j}\})\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{\unicode[STIX]{x1D707}_{\text{CL}}(d(S_{j}))p^{d(S_{j})^{2}}}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|}\mathop{\prod }_{k=1}^{d(S_{j})}(1-p^{-k})\mathop{\prod }_{k=1+d(S_{j})-h_{c}(S_{j})}^{d(S_{j})}(1-p^{-k})\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{\mathop{\prod }_{k\geqslant 1}(1-p^{-k})\mathop{\prod }_{k=1}^{d(S_{j})}(1-p^{-k})^{-2}}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|}\mathop{\prod }_{k=1}^{d(S_{j})}(1-p^{-k})\mathop{\prod }_{k=1+d(S_{j})-h_{c}(S_{j})}^{d(S_{j})}(1-p^{-k})\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|}\mathop{\prod }_{k\geqslant 1}(1-p^{-k})\mathop{\prod }_{k=1}^{d(S_{j})}(1-p^{-k})^{-1}\mathop{\prod }_{k=1+d(S_{j})-h_{c}(S_{j})}^{d(S_{j})}(1-p^{-k}).\nonumber\end{eqnarray}$$

When $S_{j}$ is $p$ -class $c$ , we have that $h_{c}(S_{j})=h(S_{j})$ , and since $S_{j}$ is a Schur $\unicode[STIX]{x1D70E}$ -ancestor, it is $Q_{c}(G)$ for some Schur $\unicode[STIX]{x1D70E}$ -group  $G$ . Since $r(G)=d(G)=d(S_{j})$ , and $r(G)\geqslant h(S_{j})$ [Reference Boston and NoverBN06, Proposition 2], we have $d(S_{j})\geqslant h_{c}(S_{j}).$ When $S_{j}$ is a Schur $\unicode[STIX]{x1D70E}$ -group, we have that $h_{c}(S_{j})=r(S_{j})=d(S_{j})$ . In either case, we conclude that

$$\begin{eqnarray}\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq S_{j}\})\geqslant \frac{1}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|}\mathop{\prod }_{k\geqslant 1}(1-p^{-k}).\end{eqnarray}$$

For all $p\geqslant 3$ , we have that $\prod _{k\geqslant 1}(1-p^{-k})>0.53$ and so

$$\begin{eqnarray}\frac{1}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|}<1.9\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq S_{j}\}).\end{eqnarray}$$

Thus,

$$\begin{eqnarray}\displaystyle \sup _{i}\mathop{\sum }_{j}a_{i,j} & = & \displaystyle \sup _{i}\mathop{\sum }_{j}\frac{|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(S_{j},S_{i})|}{|\!\operatorname{Aut}_{\unicode[STIX]{x1D70E}}(S_{j})|}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle 1.9\sup _{i}\mathop{\sum }_{j}\unicode[STIX]{x1D707}_{\text{BBH}}(\{G\mid Q_{c}(G)\simeq S_{j}\})|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(S_{j},S_{i})|\leqslant 1.9.\Box \nonumber\end{eqnarray}$$

5 Moments as an extension counting problem

Let $Q$ be a global field with a choice of place $\infty$ . (We are mainly interested in $Q=\mathbb{Q}$ or $\mathbb{F}_{q}(t)$ with the usual infinite place.) We fix a separable closure $\bar{Q}_{\infty }$ of the completion $Q_{\infty }$ . Then, inside $\bar{Q}_{\infty }$ we have the separable closure $\bar{Q}$ of $Q$ . This gives a map $\operatorname{Gal}(\bar{Q}_{\infty }/Q_{\infty })\rightarrow \operatorname{Gal}(\bar{Q}/Q)$ , and in particular distinguished decomposition and inertia groups in $\operatorname{Gal}(\bar{Q}/Q)$ at $\infty$ (as opposed to just a conjugacy classes of subgroups).

As in § 2, when $K\subset \bar{Q}$ with $K/Q$ a separable, quadratic extension, we let $K^{\operatorname{un},\infty }\subset \bar{Q}$ be the maximal extension of $K$ that is unramified everywhere and split completely at $\infty$ . We let $G_{K}^{\operatorname{un},\infty }:=\operatorname{Gal}(K^{\operatorname{un},\infty }/K)$ . We note that in $\operatorname{Gal}(K^{\operatorname{un},\infty }/Q)$ the inertia group at $\infty$ has order dividing $2$ by Lemma 2.2. Thus if $K$ is ramified at $\infty$ , we have a distinguished non-trivial inertia element $i_{K,\infty }\in \operatorname{Gal}(K^{\operatorname{un},\infty }/Q)$ . As noted earlier, an automorphism that has order dividing $2$ is called an involution. Conjugation by $i_{K,\infty }$ gives an involution of $G_{K}^{\operatorname{un},\infty }$ , and we let this conjugation be our chosen automorphism $\unicode[STIX]{x1D70E}$ of $G_{K}^{\operatorname{un},\infty }$ . (Note this is a more specific choice than we made in § 2 under different hypotheses.)

Recall, for any finite group $H$ with an involution $\unicode[STIX]{x1D70E}$ , we write $\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)$ for the continuous surjections taking conjugation by $i_{K,\infty }$ to $\unicode[STIX]{x1D70E}$ . We let $G=H\rtimes _{\unicode[STIX]{x1D70E}}C_{2}$ , and we denote the generator of $C_{2}$ by $\unicode[STIX]{x1D70E}$ (a convenient overloading of notation). Let $c$ be the set of elements of $G\setminus H$ of order  $2$ .

We define (as in [Reference Ellenberg, Venkatesh and WesterlandEVW12, § 10.2]) a marked $(G,c)$ extension of $Q$ to be $(L,\unicode[STIX]{x1D70B},m)$ such that $L/Q$ is a Galois extension of fields, $\unicode[STIX]{x1D70B}$ is an isomorphism $\unicode[STIX]{x1D70B}:\operatorname{Gal}(L/Q)\simeq G$ such that all inertia groups in $\operatorname{Gal}(L/Q)$ (except for possibly the one at $\infty$ ) have image in $\{1\}\cup c$ , and $m$ , the marking, is a homomorphism $L_{\infty }:=L\otimes _{Q}Q_{\infty }\rightarrow \bar{Q}_{\infty }$ . Note that restriction to $L$ gives a bijection between homomorphisms $L_{\infty }\rightarrow \bar{Q}_{\infty }$ and homomorphisms $L\rightarrow \bar{Q}$ . Also, note that the condition that an inertia group in $\operatorname{Gal}(L/Q)$ has image in $\{1\}\cup c$ is equivalent to requiring that it has trivial intersection with $\unicode[STIX]{x1D70B}^{-1}(H)$ because any element in $G\setminus (\{1\}\cup c)$ is either in $H$ or has square non-trivial in $H$ . Two marked $(G,c)$ extensions $(L_{1},\unicode[STIX]{x1D70B}_{1},m_{1})$ and $(L_{2},\unicode[STIX]{x1D70B}_{2},m_{2})$ are isomorphic when there is an isomorphism $L_{1}\rightarrow L_{2}$ taking $\unicode[STIX]{x1D70B}_{1}$ to $\unicode[STIX]{x1D70B}_{2}$ and $m_{1}$ to $m_{2}$ . The marking $m$ in a marked $(G,c)$ extension $(L,\unicode[STIX]{x1D70B},m)$ gives a map $\operatorname{Gal}(\bar{Q}_{\infty }/Q_{\infty })\rightarrow \operatorname{Gal}(L/Q)$ . Composing with $\unicode[STIX]{x1D70B}$ we get an infinity type $\operatorname{Gal}(\bar{Q}_{\infty }/Q_{\infty })\rightarrow G$ . Such a homomorphism is called ramified if the image of inertia is non-trivial.

Note that in each isomorphism class of marked $(G,c)$ extensions of $Q$ , there is a distinguished element such that $L\subset \bar{Q}$ and $m|_{L}$ is the inclusion map.

Theorem 5.1. Let $Q$ be a global field with a choice of place $\infty$ . Let $H$ be a finite group with involution $\unicode[STIX]{x1D70E}$ , let $G:=H\rtimes _{\unicode[STIX]{x1D70E}}C_{2}$ , and let $c$ be the set of order- $2$ elements of $G\setminus H$ . Let $\unicode[STIX]{x1D719}:\operatorname{Gal}(\bar{Q}_{\infty }/Q_{\infty })\rightarrow G$ be a ramified homomorphism with image $\langle (1,\unicode[STIX]{x1D70E})\rangle$ . There is a bijection between

$$\begin{eqnarray}\displaystyle & & \displaystyle \{(K,f)|K\subset \bar{Q},[K:Q]\nonumber\\ \displaystyle & & \displaystyle \quad =2,K_{\infty }/Q_{\infty }~\text{the quadratic extension given by}~\ker (\unicode[STIX]{x1D719}),f\in \operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)\}\nonumber\end{eqnarray}$$

and

$$\begin{eqnarray}\{\text{isomorphism classes of marked}~(G,c)~\text{extensions}~(L,\unicode[STIX]{x1D70B},m)~\text{of}~Q~\text{with infinity type}~\unicode[STIX]{x1D719}\}.\end{eqnarray}$$

In this bijection, we have $\operatorname{Disc}(L)=\operatorname{Disc}(K)^{|H|}$ .

6 Applying methods of Ellenberg–Venkatesh–Westerland to the extension counting problem

Theorem 1.5 will follow from Corollary 6.5 in this section. We will prove this result using a method and many results due to Ellenberg, Venkatesh and Westerland in papers [Reference Ellenberg, Venkatesh and WesterlandEVW16, Reference Ellenberg, Venkatesh and WesterlandEVW12]. The method counts extensions of function fields by considering this as a problem of counting $\mathbb{F}_{q}$ points on a moduli space of curves with maps to $\mathbb{P}^{1}$ , applying the Grothendieck–Lefschetz trace formula to count these points, and using results from topology to bound the dimensions of the cohomology groups.

6.1 Group theory computation

In this section, we will prove a lemma in group theory that will be central to proving Theorem 1.5. This lemma will count $\mathbb{F}_{q}$ -rational components in a moduli space on which we will eventually count points.

First we will define the universal marked central extension $\tilde{G}$ of a finite group $G$ for a union $c$ of conjugacy classes of $G$ , following [Reference Ellenberg, Venkatesh and WesterlandEVW12, § 7]. Let $C$ be a Schur cover of $G$ so we have an exact sequence

$$\begin{eqnarray}1\rightarrow H_{2}(G,\mathbb{Z})\rightarrow C\rightarrow G\rightarrow 1\end{eqnarray}$$

by the Schur covering map. For $x,y\in G$ that commute, let $\hat{x}$ and ${\hat{y}}$ be arbitrary lifts to $C$ , and let $\langle x,y\rangle$ be the commutator $[\hat{x},{\hat{y}}]\in C$ , which actually lies in $H_{2}(G,\mathbb{Z})$ since $x$ and $y$ commute. It we take the quotient of the above exact sequence by all $\langle x,y\rangle$ for $x\in c$ and $y$ commuting with $x$ , we obtain an exact sequence

$$\begin{eqnarray}1\rightarrow H_{2}(G,c)\rightarrow \tilde{G}_{c}\rightarrow G\rightarrow 1,\end{eqnarray}$$

which is still a central extension. Let $G^{\text{ab}}$ denote the abelianization of $H$ . The universal marked central extension is $\tilde{G}=\tilde{G}_{c}\times _{G^{\text{ab}}}\mathbb{Z}^{c/G}$ , where $c/G$ denotes the set of conjugacy classes in $c$ and the map $\mathbb{Z}^{c/G}\rightarrow G^{\text{ab}}$ sends each standard generator to an element of the associated conjugacy class. We have a map $\tilde{G}\rightarrow G$ , given through projecting to the first factor. (See [Reference Ellenberg, Venkatesh and WesterlandEVW12, § 7] for why this is called a universal marked central extension.)

Lemma 6.1. Let $H$ be an odd finite group with a GI-automorphism $\unicode[STIX]{x1D70E}$ , and $G=H\rtimes _{\unicode[STIX]{x1D70E}}C_{2}$ . Let $c$ be the (single) conjugacy class of order- $2$ elements. Let $q$ be a power of a prime and $n$ be an odd integer. If $(q,2|H|)=1$ and $(q-1,|H|)=1$ , then for each $y\in c$ , there is exactly $1$ element $x\in \tilde{G}_{c}$ such that $(x,n)\in \tilde{G}$ , and $x$ has image $y$ in $G$ , and $x^{q}=x$ .

Proof. We have that $|\tilde{G}_{c}|=2|H||H_{2}(G,c)|$ and that $H_{2}(G,\mathbb{Z})$ is a quotient of $H_{2}(H,\mathbb{Z})$ by [Reference Ellenberg, Venkatesh and WesterlandEVW12, Example 9.3.2]. Thus since $|H|$ is relatively prime to $2(q-1)$ , we have that $|H_{2}(G,\mathbb{Z})|$ is as well and thus $|H_{2}(G,c)|$ is as well. Since $|\tilde{G}_{c}|/2$ is relatively prime to $q-1$ , we have that for $x\in \tilde{G}_{c}$ , $x^{q}=x$ if and only if $x^{2}=1$ .

Let $w\in \tilde{G}_{c}$ be in the inverse image of $y$ . Then we ask for which $k\in H_{2}(G,c)$ is $wk$ of order  $2$ . Since $H_{2}(G,c)\rightarrow \tilde{G}_{c}$ is central, we have $(wk)^{2}=w^{2}k^{2}$ , and note $w^{2}\in H_{2}(G,c)$ since $y^{2}=1$ . Since $H_{2}(G,c)$ is an odd abelian group, there is exactly one $k\in H_{2}(G,c)$ such that $w^{2}k^{2}=1$ . Let $x=wk$ for this $k$ , which is the only possible $x$ satisfying the conditions of the lemma. Also, note that $(x,n)\in \tilde{G}$ since $x$ and $n$ have image of the class of $y$ in $G^{\text{ab}}$ , proving the lemma.◻

6.3 Counting $\mathbb{F}_{q}$ points

In this section, we will count the $\mathbb{F}_{q}$ points of $\mathsf{CHur}^{c,c}$ in Theorem 6.4, and then use our Theorem 5.1 to translate that into a result about surjections from Galois groups $G_{K}$ in Corollary 6.5, which will finally prove Theorem 1.5.

Theorem 6.4. Given $G$ and $c$ as in Theorem 6.2, we have a constant $C$ and a constant $n_{G}$ such that for $q>C^{2}$ , with $(q,|G|)=1$ and $(q-1,|G|/2)=1$ , and odd $n\geqslant n_{G}$ ,

$$\begin{eqnarray}|\#\mathsf{CHur}_{G,n}^{c,c}(\mathbb{F}_{q})-q^{n}\cdot \#c|\leqslant \frac{q^{n}}{\sqrt{q}/C-1}.\end{eqnarray}$$

Proof. Our theorem will follow by applying the Grothendieck–Lefschetz trace formula to $X:=\mathsf{CHur}_{G,n}^{c,c}\otimes _{\mathbb{Z}[|G|^{-1}]}\mathbb{F}_{q}$ . By Theorem 6.2(i), we have that $X$ is smooth of dimension $n$ . We have that $\dim H_{\text{c},\acute{\text{e}}\text{t}}^{i}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })=\dim H_{\acute{\text{e}}\text{t}}^{2n-i}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })$ by Poincaré duality.

Next, we will relate $\dim H_{\acute{\text{e}}\text{t}}^{j}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })$ to $\dim H^{j}(\mathsf{CHur}_{G,n}^{c,c}(\mathbb{C}),\mathbb{Q}_{\ell })$ for some $\ell >n$ . To compare étale cohomology between characteristic 0 and positive characteristic, we will use [Reference Ellenberg, Venkatesh and WesterlandEVW16, Proposition 7.7]. The result [Reference Ellenberg, Venkatesh and WesterlandEVW16, Proposition 7.7] gives an isomorphism between étale cohomology between characteristic 0 and positive characteristic in the case of a finite cover of a complement of a reduced normal crossing divisor in a smooth proper scheme. Though [Reference Ellenberg, Venkatesh and WesterlandEVW16, Proposition 7.7] is only stated for étale cohomology with coefficients in $\mathbb{Z}/\ell \mathbb{Z}$ , the argument goes through identically for coefficients in $\mathbb{Z}/\ell ^{k}\mathbb{Z}$ , and then we can take the indirect limit and tensor with $\mathbb{Q}_{\ell }$ to obtain the result of [Reference Ellenberg, Venkatesh and WesterlandEVW16, Proposition 7.7] with $\mathbb{Z}/\ell \mathbb{Z}$ coefficients replaced by $\mathbb{Q}_{\ell }$ coefficients. So we apply this strengthened version to conclude that $\dim H_{\acute{\text{e}}\text{t}}^{j}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })=\dim H_{\acute{\text{e}}\text{t}}^{j}((\mathsf{CHur}_{G,n}^{c,c})_{\mathbb{C}},\mathbb{Q}_{\ell })$ . (As in [Reference Ellenberg, Venkatesh and WesterlandEVW16, proof of Proposition 7.8], we apply comparison to $\mathsf{CHur}_{G,n}^{c,c}\times _{\operatorname{Conf}^{n}}\text{PConf}_{n}$ , where $\text{PConf}_{n}$ is the moduli space of $n$ labeled points on $\mathbb{A}^{1}$ and is the complement of a relative normal crossings divisor in a smooth proper scheme [Reference Ellenberg, Venkatesh and WesterlandEVW16, Lemma 7.6]. Then we take $S_{n}$ invariants to compare the étale cohomology of $\mathsf{CHur}_{G,n}^{c,c}$ across characteristics.) By the comparison of étale and analytic cohomology [Reference Artin, Grothendieck and VerdierSGA4(3), Exposé XI, Theorem 4.4] $\dim H^{j}(\mathsf{CHur}_{G,n}^{c,c}(\mathbb{C}),\mathbb{Q}_{\ell })=\dim H_{\acute{\text{e}}\text{t}}^{j}((\mathsf{CHur}_{G,n}^{c,c})_{\mathbb{C}},\mathbb{Q}_{\ell })$ .

By Theorem 6.2(iii), there is a constant $C$ such that for all $j\geqslant 1$ and for all $n$ , we have $\dim H^{j}(\mathsf{CHur}_{G,n}^{c,c}(\mathbb{C}),\mathbb{Q}_{\ell })\leqslant C^{j}$ . Thus $\dim H_{\acute{\text{e}}\text{t}}^{j}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })\leqslant C^{j}$ for all $j\geqslant 1$ . Thus using Poincaré duality, $\dim H_{\acute{\text{e}}\text{t},c}^{i}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })\leqslant C^{2n-i}$ for all $i<2n$ . By Theorem 6.2(iv) and Lemma 6.1, we have that $X$ has $\#c$ components fixed by $\operatorname{Frob}$ for odd $n\geqslant n_{G}$ for some fixed  $n_{G}$ .

Then by the Grothendieck–Lefschetz trace formula we have

$$\begin{eqnarray}\#X(\mathbb{F}_{q})=\mathop{\sum }_{j\geqslant 0}(-1)^{j}\operatorname{Tr}(\operatorname{Frob}\!|_{H_{\text{c},\acute{\text{e}}\text{t}}^{j}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })})\end{eqnarray}$$

and also we know $\operatorname{Tr}(\operatorname{Frob}\!|_{H_{\text{c},\acute{\text{e}}\text{t}}^{2n}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })})$ is $q^{n}$ times the number of components of $X$ fixed by $\operatorname{Frob}$ . Since $X$ is smooth, we have that the absolute value of any eigenvalue of $\operatorname{Frob}$ on $H_{\text{c},\acute{\text{e}}\text{t}}^{j}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })$ is at most $q^{j/2}$ . Thus, for odd $n\geqslant n_{G}$ ,

$$\begin{eqnarray}\displaystyle |\#X(\mathbb{F}_{q})-q^{n}\times \#c| & = & \displaystyle \biggl|\mathop{\sum }_{0\leqslant j<2\dim X}(-1)^{j}\operatorname{Tr}(\operatorname{Frob}\!|_{H_{\text{c},\acute{\text{e}}\text{t}}^{j}(X_{\bar{\mathbb{F}}_{q}},\mathbb{Q}_{\ell })})\biggr|\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \mathop{\sum }_{0\leqslant j<2\dim X}q^{j/2}C^{2n-j}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle q^{n}\mathop{\sum }_{1\leqslant i}(\sqrt{q}/C)^{-i}.\nonumber\end{eqnarray}$$

The theorem follows. ◻

We have $Q=\mathbb{F}_{q}(t)$ and $Q_{\infty }=\mathbb{F}_{q}((t^{-1}))$ , for $q$ odd. Unlike in the number field case, in which there is only one possible ramified quadratic extension of $\mathbb{Q}_{\infty }=\mathbb{R}$ , here there are two ramified quadratic extensions of $Q_{\infty }=\mathbb{F}_{q}((t^{-1}))$ . If $K/\mathbb{F}_{q}(t)$ is a quadratic extension, we say it is imaginary quadratic of type I if $K_{\infty }\simeq \mathbb{F}_{q}((t^{-1/2}))$ and of type II if $K_{\infty }\simeq \mathbb{F}_{q}(((\unicode[STIX]{x1D6FC}t)^{-1/2}))$ for an $\unicode[STIX]{x1D6FC}\in \mathbb{F}_{q}\setminus \mathbb{F}_{q}^{2}$ . Let $IQ_{n}$ be the set of $K\subset \bar{Q}$ such that $K$ is imaginary quadratic of type I and $\operatorname{Nm}\operatorname{Disc}(K)=q^{n+1}$ . Let $IQ_{n}^{\prime }$ be the set of $K\subset \bar{Q}$ such that $K$ is imaginary quadratic of type II and $\operatorname{Nm}\operatorname{Disc}(K)=q^{n+1}$ .

Corollary 6.5. Let $H$ be an odd finite group with GI-automorphism $\unicode[STIX]{x1D70E}$ such that $H\rtimes _{\unicode[STIX]{x1D70E}}C_{2}$ is center-free. As $q$ ranges through powers of primes such that $(q,2|H|)=1$ and $(q-1,|H|)=1$ , we have

$$\begin{eqnarray}\lim _{q\rightarrow \infty }\limsup _{\substack{ n\rightarrow \infty \\ n~\text{odd}}}\frac{\mathop{\sum }_{K\in IQ_{n}}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)|}{\#IQ_{n}}=1.\end{eqnarray}$$

The same result holds if we replace $\limsup$ by $\liminf$ and/or replace $IQ_{n}$ by $IQ_{n}^{\prime }$ .

Theorem 1.5 then follows from Corollary 6.5 after noting that $H\rtimes _{\unicode[STIX]{x1D70E}}C_{2}$ is center-free if and only if the center of $H$ contains no elements fixed by $\unicode[STIX]{x1D70E}$ except the identity.

Proof. By Theorem 6.2(ii) the points $\mathsf{CHur}_{G,n}^{c,c}(\mathbb{F}_{q})$ are in bijection with isomorphism classes of marked $(G,c)$ extensions $(L,\unicode[STIX]{x1D70B},m)$ of $Q$ with certain infinity types  $\unicode[STIX]{x1D719}$ . These infinity types are all $G$ -conjugate, and there are $\#c$ of them. Let $\unicode[STIX]{x1D719}_{0}$ be the infinity type such that $\unicode[STIX]{x1D719}(F_{\unicode[STIX]{x1D6E5}})=1$ and $\operatorname{im}\unicode[STIX]{x1D719}=\langle (1,\unicode[STIX]{x1D70E})\rangle$ . Note that $\mathbb{F}_{q}((t^{-1/2}))$ is the imaginary quadratic extension given by $\ker (\unicode[STIX]{x1D719}_{0})$ .

Let $\unicode[STIX]{x1D719}:\operatorname{Gal}(\bar{Q}_{\infty }/Q_{\infty })\rightarrow G$ be a ramified homomorphism with image $\langle (1,\unicode[STIX]{x1D70E})\rangle$ , let $g\in G$ , and let $\unicode[STIX]{x1D719}^{g}$ denote the conjugation. Then isomorphism classes of marked $(G,c)$ extensions $(L,\unicode[STIX]{x1D70B},m)$ of $Q$ with infinity type $\unicode[STIX]{x1D719}$ of a given discriminant are in bijection with isomorphism classes of marked $(G,c)$ extensions $(L,\unicode[STIX]{x1D70B},m)$ of $Q$ with infinity type $\unicode[STIX]{x1D719}^{g}$ and that discriminant by sending $(L,\unicode[STIX]{x1D70B},m)$ to $(L,\unicode[STIX]{x1D70B}^{g},m)$ . So, we have that

$$\begin{eqnarray}\displaystyle \#\mathsf{CHur}_{G,n}^{c,c}(\mathbb{F}_{q}) & = & \displaystyle \#c\cdot \#\{\text{isomorphism classes of marked}~(G,c)\text{-}\text{extensions}~L/\mathbb{F}_{q}(t)~\text{of infinity}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \hspace{5.0pt}\text{type}~\unicode[STIX]{x1D719}_{0}\text{ and }\operatorname{Nm}\operatorname{Disc}(L)=q^{(n+1)|H|}\}.\nonumber\end{eqnarray}$$

Further, by Theorem 5.1, we then conclude that

$$\begin{eqnarray}\displaystyle \#\mathsf{CHur}_{G,n}^{c,c}(\mathbb{F}_{q}) & = & \displaystyle \#c\cdot \{(K,f)\mid K\subset \bar{Q},K~\text{imaginary quadratic type I},\nonumber\\ \displaystyle & & \displaystyle \qquad \hspace{6.00006pt}f\in \operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H),\operatorname{Nm}\operatorname{Disc}(K)=q^{n+1}\}.\nonumber\end{eqnarray}$$

So by Theorem 6.4, we have a constant $C$ , only depending on $H$ , such that for $q\geqslant 4C^{2}$ and odd $n\geqslant n_{G}$

$$\begin{eqnarray}\biggl|\mathop{\sum }_{K\in IQ_{n}}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)|-q^{n}\biggr|\leqslant 2Cq^{n-1/2}.\end{eqnarray}$$

Thus, for $q\geqslant 4C^{2}$ and all odd $n\geqslant n_{G}$

$$\begin{eqnarray}\frac{\mathop{\sum }_{K\in IQ_{n}}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)|}{\#IQ_{n}}=\frac{q^{n}+O(q^{n-1/2})}{q^{n}-q^{n-1}}=1+O(q^{-1/2}).\end{eqnarray}$$

It follows that the limit as $q\rightarrow \infty$ , of the of $\limsup$ or $\liminf$ , in odd $n$ , of the lefthand side are both 1. For the case of $IQ_{n}^{\prime }$ , we have a bijection $K\mapsto K\otimes _{\mathbb{F}_{q}(t)}\mathbb{F}_{q}(t)$ (where the map $\mathbb{F}_{q}(t)\rightarrow \mathbb{F}_{q}(t)$ is given by $t\mapsto \unicode[STIX]{x1D6FC}t$ , for some $\unicode[STIX]{x1D6FC}\in \mathbb{F}_{q}\setminus \mathbb{F}_{q}^{2}$ ) between $IQ_{n}$ and $IQ_{n}^{\prime }$ that preserves $G_{K}^{\operatorname{un},\infty }$ .◻

6.4 Further results assuming a conjecture on the homology of Hurwitz spaces

The program developed by Ellenberg, Venkatesh and Westerland in [Reference Ellenberg, Venkatesh and WesterlandEVW12] aims to prove stronger results on the topology of Hurwitz spaces, from which corresponding stronger results on the point counts would follow. For example, $\text{HS}_{\unicode[STIX]{x1D6FC}}$ [Reference Ellenberg, Venkatesh and WesterlandEVW12, § 11.1] is a conjecture on the homology of Hurwitz spaces for a given group $G$ and conjugacy invariant subset  $c$ .

Theorem 6.6. Let $H$ be an odd finite group with GI-automorphism $\unicode[STIX]{x1D70E}$ such that $H\rtimes _{\unicode[STIX]{x1D70E}}C_{2}$ is center-free. If $\text{HS}_{\unicode[STIX]{x1D6FC}}$ holds for $G=H\rtimes _{\unicode[STIX]{x1D70E}}C_{2}$ and $c$ the order- $2$ elements of $G$ , then there is a $q_{0}$ such that for $q\geqslant q_{0}$ , with $(q,2|H|)=1$ and $(q-1,|H|)=1$ , we have

$$\begin{eqnarray}\limsup _{\substack{ n\rightarrow \infty \\ n~\text{odd}}}\frac{\mathop{\sum }_{K\in IQ_{n}}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)|}{\#IQ_{n}}=1.\end{eqnarray}$$

The same result holds if we replace $IQ_{n}$ by $IQ_{n}^{\prime }$ .

Proof. We apply Theorem 5.1 and [Reference Ellenberg, Venkatesh and WesterlandEVW12, Theorem 11.1.1]. Lemma 6.1 shows that the quantity $B(L_{\infty },\mathfrak{m})$ appearing in [Reference Ellenberg, Venkatesh and WesterlandEVW12, Theorem 11.1.1] is 1. Finally, we use that an étale $G$ -extension $L_{\infty }$ has $|G|/|\!\operatorname{Aut}_{G}(L_{\infty })|$ corresponding infinity types and a $G$ -extension has $|G|$ markings.◻

7 Non-equivariant moments

While in this paper, we have asked about the equivariant moments, or averages of $|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G_{K}^{\operatorname{un},\infty },H)|$ , one could naturally ask about non-equivariant moments, or averages of  $|\!\operatorname{Sur}(G_{K}^{\operatorname{un},\infty },H)|$ . It turns out these non-equivariant moments reduce in a simple way to equivariant moments.

Let $G$ be a group with a GI-automorphism $\unicode[STIX]{x1D70E}$ . Then we have an injection

$$\begin{eqnarray}\displaystyle \operatorname{Sur}(G,H) & \rightarrow & \displaystyle \operatorname{Hom}_{\unicode[STIX]{x1D70E}}(G,H\times H)\nonumber\\ \displaystyle f & \mapsto & \displaystyle f\times f\unicode[STIX]{x1D70E},\nonumber\end{eqnarray}$$

where the automorphism $\unicode[STIX]{x1D70E}$ of $H\times H$ is switching the factors. In fact, this is a bijection onto the subset of $\operatorname{Hom}_{\unicode[STIX]{x1D70E}}(G,H\times H)$ that surject onto the first factor. Let ${\mathcal{F}}$ be the set of $\unicode[STIX]{x1D70E}$ -invariant subgroups of $H\times H$ that surject onto the first factor. Then

(3) $$\begin{eqnarray}|\!\operatorname{Sur}(G,H)|=\mathop{\sum }_{F\in {\mathcal{F}}}|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G,F)|.\end{eqnarray}$$

Note since $\unicode[STIX]{x1D70E}$ is GI on $G$ , if it is not $GI$ on $F$ , then $|\!\operatorname{Sur}_{\unicode[STIX]{x1D70E}}(G,F)|=0$ . Thus (3) would still hold if we restrict the sum on the right to $F$ such that switching factors in $H\times H$ is $GI$ on $F$ (i.e.  $F$ generated by elements of the form $(h,h^{-1})$ for $h\in H$ ).